For soluble groups, De Falco, de Giovanni and Musella (2014) proved a sharp dichotomy: a soluble group satisfying the minimal condition on normal subgroups of infinite rank (Min-n∞n^{\infty}) either has finite rank or satisfies the full minimal condition on normal subgroups (Min-nn). Whether this extends to locally soluble groups is open. The extension is obstructed because the soluble proof rests on Baer's theorem that a soluble group with Min-nn is periodic, and this fails one class up: Heineken and Wilson (1974) constructed non-trivial torsion-free locally soluble groups satisfying Min-nn. Only a partial extension is known, for locally soluble groups whose chief factors all have finite rank. The problem is to determine whether every locally soluble group satisfying Min-n∞n^{\infty} has finite rank or satisfies Min-nn - equivalently, whether Min-n∞n^{\infty} implies Min-nn for locally soluble groups of infinite rank - by giving a proof or an explicit counterexample.

Background

In the theory of infinite groups, chain conditions play the role that finiteness plays for finite groups. The minimal condition on normal subgroups (Min-nn) says that every non-empty set of normal subgroups of a group GG has a minimal element, equivalently that there is no infinite strictly descending chain N1▹N2▹⋯N_1 \triangleright N_2 \triangleright \cdots of normal subgroups of GG. A classical theorem of Baer (1964), obtained from his study of irreducible groups of automorphisms of abelian groups (a group of automorphisms of an abelian group AA is irreducible if 00 and AA are the only invariant subgroups), states that a soluble group - a group whose derived series (the iterated commutator subgroups G≥G′≥G′′≥⋯G \geq G' \geq G'' \geq \cdots) terminates at the identity after finitely many steps - satisfying Min-nn is periodic, meaning every element has finite order. The structure theory of soluble Min-nn groups was then developed by McDougall (1970) and others, and Hartley (1977) constructed uncountable, hence infinite-rank, soluble groups satisfying Min-nn, so Min-nn by no means forces smallness.

Here rank is Prüfer (or special) rank: GG has finite rank rr if every finitely generated subgroup of GG can be generated by at most rr elements, and infinite rank otherwise. A classical theme in this area, standard since the 1950s–60s (see the survey by de Giovanni, 2014), is that in many generalized-soluble groups of infinite rank the large subgroups control the whole group: for instance, locally nilpotent groups (groups in which every finitely generated subgroup is nilpotent, i.e. has a finite central series) and locally finite groups (groups in which every finitely generated subgroup is finite) of infinite rank always contain abelian subgroups of infinite rank. This motivated an extensive research program, carried out chiefly by De Falco, de Giovanni, Musella, Sysak and Trabelsi, in which a condition normally imposed on all subgroups, or all normal subgroups, is weakened to the subgroups of infinite rank, and one asks whether the global condition is nevertheless recovered.

De Falco, de Giovanni and Musella (2014) treated the rank-weakened form of Min-nn: a group satisfies the minimal condition on normal subgroups of infinite rank, written Min-n∞n^{\infty}, if it has no infinite strictly descending chain of normal subgroups each of which has infinite rank. Since a group of finite rank has no subgroups of infinite rank at all, Min-n∞n^{\infty} holds vacuously for such groups (for example the additive group of rationals), so Min-n∞n^{\infty} alone can never imply Min-nn without some rank hypothesis. The main theorem of that paper is the sharp dichotomy for soluble groups: if a soluble group GG satisfies Min-n∞n^{\infty}, then GG has finite rank or GG satisfies Min-nn; in particular, for soluble groups of infinite rank, Min-n∞n^{\infty} implies Min-nn. The proof leans essentially on Baer's periodicity: it first shows that a torsion-free abelian normal subgroup of a soluble Min-n∞n^{\infty} group must have finite rank, and torsion is what makes the soluble machinery run.

For locally soluble groups - groups in which every finitely generated subgroup is soluble, with no bound on the derived length - both pillars of the soluble proof fail. Heineken and Wilson (1974) constructed non-trivial locally soluble torsion-free groups satisfying the full Min-nn (using iterated twisted-wreath-type constructions they call treble products), so Baer's periodicity theorem is false one class up; and there is no finite derived length on which to induct. De Falco, de Giovanni and Musella obtained a partial extension: a locally soluble group satisfying Min-n∞n^{\infty} all of whose chief factors have finite rank must itself have finite rank. Here a chief factor is a minimal normal section H/KH/K, where K<HK < H are normal in GG and H/KH/K is a minimal non-trivial normal subgroup of G/KG/K. What remains untouched is the locally soluble case without restrictions on the chief factors, and this is the present problem.

Problem Statement

Let GG be a locally soluble group (every finitely generated subgroup of GG is soluble) satisfying the minimal condition on normal subgroups of infinite rank: there is no infinite strictly descending chain

N1▹N2▹N3▹⋯N_1 \triangleright N_2 \triangleright N_3 \triangleright \cdots

of normal subgroups of GG such that each NiN_i has infinite Prüfer rank. Must GG either have finite Prüfer rank or satisfy the full minimal condition on normal subgroups (Min-nn)? Equivalently: if GG is locally soluble, of infinite rank, and satisfies Min-n∞n^{\infty}, does GG satisfy Min-nn?

Resolve this by a proof, or by an explicit counterexample - a locally soluble group of infinite rank satisfying Min-n∞n^{\infty} but not Min-nn. This is the question left open by De Falco, de Giovanni and Musella (2014), who settled the soluble case; the locally soluble case is open even though a partial result is known for locally soluble groups all of whose chief factors have finite rank.

References

  1. M. De Falco, F. de Giovanni, C. Musella, 'A note on soluble groups with the minimal condition on normal subgroups', Journal of Algebra and Its Applications 13 (2014), no. 4, article 1350134. DOI: 10.1142/S021949881350134X, https://doi.org/10.1142/S021949881350134X (source of the open question)
  2. R. Baer, 'Irreducible groups of automorphisms of abelian groups', Pacific Journal of Mathematics 14 (1964), 385-406. DOI: 10.2140/pjm.1964.14.385, https://msp.org/pjm/1964/14-2/p02.xhtml
  3. H. Heineken, J. S. Wilson, 'Locally soluble groups with min-n', Journal of the Australian Mathematical Society 17 (1974), 305-318. DOI: 10.1017/S1446788700017079, https://doi.org/10.1017/S1446788700017079
  4. B. Hartley, 'Uncountable Artinian modules and uncountable soluble groups satisfying Min-n', Proceedings of the London Mathematical Society (3) 35 (1977), 55-75. DOI: 10.1112/plms/s3-35.1.55, https://doi.org/10.1112/plms/s3-35.1.55
  5. D. McDougall, 'Soluble groups with the minimum condition for normal subgroups', Mathematische Zeitschrift 118 (1970), 157-167. DOI: 10.1007/BF01113337, https://doi.org/10.1007/BF01113337
  6. M. De Falco, F. de Giovanni, C. Musella, 'Groups with finitely many conjugacy classes of non-normal subgroups of infinite rank', Colloquium Mathematicum 131 (2013), 233-239. DOI: 10.4064/cm131-2-6, https://doi.org/10.4064/cm131-2-6
  7. M. De Falco, F. de Giovanni, C. Musella, 'Large soluble groups and the control of embedding properties', Ricerche di Matematica 63 (2014), 117-130. DOI: 10.1007/s11587-014-0204-x, https://link.springer.com/article/10.1007/s11587-014-0204-x
  8. M. De Falco, F. de Giovanni, C. Musella, N. Trabelsi, 'Groups of infinite rank with finite conjugacy classes of subnormal subgroups', Journal of Algebra 431 (2015), 24-37. DOI: 10.1016/j.jalgebra.2015.01.031, https://doi.org/10.1016/j.jalgebra.2015.01.031
  9. F. de Giovanni, 'Infinite groups with rank restrictions on subgroups', Journal of Mathematical Sciences 199 (2014), 261-265. DOI: 10.1007/s10958-014-1854-7, https://link.springer.com/article/10.1007/s10958-014-1854-7